, giving r = 25 pc.
Main sequence fitting can be used out to distances of about 7 kpc, still not reaching out of our galaxy. We now see why we use the phrase ’cosmic distance ladder’. The parallax method reaches out to about 1000 pc.
After that, main sequence fitting needs to be used. But in order to use main sequence fitting, we needed a calibratet HR-diagram like figure 3.
But in order to obtain such a diagram, the parallax method needed to be used on nearby clusters. So we need to go step by step, first the parallax method which we use to calibrate the HR-diagram to be used for the main sequence fitting at larger distances. Now we will continue one more step up the ladder. We use stars in clusters which distance is calibrated with main sequence fitting in order to calibrate the distance indicators to be used for larger distances.
3 Distance indicators
Again the method is based on equation (2). We can always measure the apparent magnitude m of a distant object. From the equation, we see that all we need in order to obtain the distance is the absolute magnitude. If
Figure 4: The HR-diagrams for the example exercise (note: spectral class is just a different measure of temperature). The upper plot shows the HR-diagram of a cluster with a known distance. Since the distance is known, we have been able to convert the apparent magnitudes to absolute magnitudes and we therefore plot absolute magnitudes on the y-axis. The lower plot is the HR-diagram of a cluster with unknown distance. Because of the unknown distance, we only have information about the apparent magnitude of the stars and therefore we now have apparent magnitude on the y-axis.
we know the absolute magnitude (luminosity) for an object, we can find its distance. But how do we know the absolute magnitude? There are a few classes of objects, called standard candles, which reveal their absolute magnitude in different ways. Examples of these ’standard candles’ can be Cepheid stars or supernova explosions.
Another class of distance indicators are the so-called ’standard rulers’. The basis for the distance determination with standard rulers is the small-angle formula,
d = θr,
where d is the physical length of an object, r is the distance to the object and θ is the apparent angular extension (length) of the object. We can often measure the angular extension of an observed object. All that we need in order to find the distance is the physical length d. There are some objects for which we know the physical length. These objects are called standard rulers. For instance a special kind of galaxy which has been shown to always have the same dimensions could be used as a standard ruler.
3.1 Cepheid stars as distance indicators
Several stars show periodic changes in their apparent magnitudes. This was first thought to be caused by dark spots on a rotating star’s surface:
When the dark spots were turned towards us, the star appeared fainter, when the spots were turned away from us, the star appeared brighter. To-day we know that these periodic variations in the star’s magnitude is due to pulsations. The star is pulsating and therefore periodically changing its radius and surface temperature.
The Milky Way has two small satellite galaxies orbiting it, the Large and the Small Magellanic Cloud (LMC and SMC). They contain 109 − 1010 stars, less than one tenth of the number of stars in the Milky Way and are located at a distance of about 160 000 ly (LMC) and 200 000 ly (SMC) from the Sun. In 1908, Henrietta Leavitt at Harvard University discovered about 2400 of these pulsating stars in the SMC. The pulsation period of these stars were found to be in the range between 1 and 50 days. These stars were called Cepheids named after one of the first pulsating stars to be discovered, δ Cephei. She found a relationship between the stars’ apparent magnitude and pulsation period. The shorter/longer the pulsation period, the fainter/brighter the star. Since all these stars were in the SMC they were all at roughly the same distance to us. We have seen above that for stars at the same distance, there is a constant difference M − m in apparent and absolute magnitude. So the stars with a larger/smaller apparent magnitude also had a larger/smaller absolute magnitude. Since absolute magnitude is a measure of luminosity, what she had found was a period-luminosity relation.
Pulsating stars with higher luminosity were thus found to be pulsating with longer periods, pulsating stars with low luminosity were found to be pulsating with short periods. We can now reverse the argument: By mea-suring the period one can obtain the luminosity. There was one problem however: The method could not be calibrated as the distance to the SMC was unknown and therefore also the constant in m − M = constant was unknown. Without this constant one cannot find M . One had to find Cepheids in our vicinity for which the distance was known. Only in that way could this constant and thus the relation between period and absolute magnitude be established.
Today the distance to several Cepheids in our galaxy are known by other methods. One of the most recent measurements of the constants in the period-luminosity relation came from the parallax measurements of several Cepheids by the Hipparcos satellite. The relation was found to be
MV = −2.81 log10Pd− 1.43,
where Pd is the period in days. Here MV is the absolute magnitude in the Visual part of the electromagnetic spectrum instead of the normal magnutide M which is based on the flux integrated over all wavelengths λ. Before describing in detail the difference between M and MV, we will end our discussion on the Cepheid stars.
When pulsating stars were first used to measure distances one did not know that there are three different types of pulsating stars with different period-luminosity relations:
1. The classical Cepheids which belong to a class of giants, are very luminous stars. These are the most useful distance indicators for large distances because of their high luminosity.
2. W Virginis stars, or type II Cepheids are pulsating stars which on average have lower luminosity than the classical Cepheids.
3. RR Lyrae stars are small stars which usually have less mass than the Sun. Their luminosity is much lower than the luminosity of classical Cepheids and RR Lyrae stars are therfore less useful for distance determination at large distances. The advantage with RR Lyrae stars however, is that they are much more numerous than classical Cepheids.
When Edwin Hubble tried to estimate the distance to our neighbour galaxy Andromeda, he obtained a distance of about one million light years whereas the real distance is about twice as large. The reason for this er-ror was that he observed W Virginis stars in Andromeda and applied the period-luminosity relation for classical Cepheids, thinking that they were the same. In this course we will mainly discuss the classical Cepheids.
Since Cepheids are very lumious (about 103to 104 times higher luminosity than the Sun) they can be observed in distant galaxies. In order to
de-termine the distance of a whole galaxy it suffices to find Cepheid stars in that galaxy and determine their distance. In this manner, the distance to several galaxies out to about 30 Mpc has been measured. Beyond 30 Mpc other methods need to be applied.
At the moment we will use the period-luminosity relation for Cepheids to determine distances without questioning why it works. When we come to the lectures on stellar structure we will study the physics behind these pulsations and see if we can deduce the period-luminosity relation theo-retically by doing physics in the interior of stars.
We have now learned about our first distance indicator: We can find the absolute magnitude MV at visual wavelength of Cepheids by observing their plusation period. Having the absolute magnitude MV we can find the distance. We will now look at a different approach to find MV for a distant object, but first we will discuss some extended definitions of magnitudes.
3.2 Magnitudes and color indices
Looking back at the definition of absolute magnitude, we see that we can write the absolute magnitude M as
M = Mref − 2.5 log10 where Mref and Fref are the absolute magnitude and flux (observed flux if the distance had been 10 pc) of a reference star used for calibration (as we have seen before, the star Vega with its magnitude defined to be 0, has often been used for this purpose). The flux is here the total flux of the star integrated over all wavelengths
F = Z ∞
0
F (λ)dλ. (3)
The magnitude M which is based on flux integrated over all wavelengths is called the bolometric magnitude.
The visual magnitude MV on the other hand, is based on the flux over a wavelength region defined by a filter function SV(λ). The filter function is a function which is centered at λ = 550 nm with an effective bandwidth of 89 nm. The flux FV which is used instead of F to define visual magnitude can be written as
FV = Z ∞
0
F (λ)SV(λ)dλ.
Compare with expression (3): The main difference is that a limited wave-length range is selected by SV(λ). The magnitude is then defined as
MV = MVref − 2.5 log10 FV FVref
.
As for the bolometric magnitude, the relation between absolute and ap-parent visual magnitude is also given by
MV − mV = −5 log10
r
10 pc
.
The concept of the visual magnitude originates from the fact that detectors normally do not observe the flux over all wavelengths. Instead detectors are centered on a given wavelength and integrate over wavelengths around this center wavelength in a given bandwidth. There are three of these filters which are in common use:
• U-filter (ultraviolet), λ0 = 365 nm, ∆λF W HM = 68 nm
• B-filter (blue), λ0 = 440 nm, ∆λF W HM = 98 nm
• V-filter (visual), λ0 = 550 nm, ∆λF W HM = 89 nm
The absolute magnitudes MV, MBand MU are used to define color indices.
These color indices (U − B) and (B − V ) are defined as U − B = MU − MB = mU− mB, B − V = MB− MV = mB− mV.
Note that these indices are written as a difference in apparent or absolute magnitudes: The color indices are independent of distance and will there-fore give the same results if they are obtained using apparent magnitudes or absolute magnitudes (check that you can show this mathematically!).
These indices are used to measure several properties of a star related to its color. The period-luminosity relation for a Cepheid can be improved using information about its color in terms of the (B − V ) color index as
MV = −3.53 log10Pd− 2.13 + 2.13(B − V ).
For Cepheids, the B−V color index is usually in the range 0.4 to 1.1. Thus, a more exact MV and thereby a more exact distance (using relation (2)) can be obtained using the additional distance independent information contained in the color of the star. It suffices to observe the star with three color filters instead of one to obtain this additional information.
3.3 Supernovae as distance indicators
One of the most energetic events in the Universe are the Supernova ex-plosions. In such an explosion, one star might emit more energy than the total energy emitted by all the stars in a galaxy. For this reason, super-nova explosions can be seen at very large distances. The last confirmed supernova in the Milky way was seen in 1604 and was studied by Kepler.
It reached an appaerent magnitude of about −2.5, similar to Jupiter at its brightest. There have been other reports of supernovae in the Milky
way during the last 2000–3000 years, both in Europe and Asia. Some of these were so bright that they were seen clearly in the sky during daylight.
Written material from Europe, Asia and the middle East all report about a supernova in 1006 which was so bright that one could use it to read at night time. The nearest supernova in modern times, called SN1987A, was observed in 1987 in the Large Magellanic Cloud at a distance of 51 kpc.
It was visible by the naked eye from the southern hemisphere.
Supernovae can be classified as type I or type II,
1. Type I supernovae: These explosions show no hydrogen lines. There are three sub types, defined according to their spectra: Type Ia, Ib and Ic.
2. Type II supernovae: These are explosions with strong hydrogen lines. Type II supernova have several properties in common with type Ib and Ic.
It is now clear that supernovae of type Ib, Ic and II are core collapse supernovae. This is a star ending its life in a huge explosion, leaving behind a neutron star or a black hole. In the lectures on stellar evolution we will come back to the details of a core collapse supernova. Type Ia supernovae are usually brighter. These have the property which is desirable for a standard candle: Their luminosity is relatively constant and there is a receipe for finding their exact luminosity. The origin of type Ia supernovae are still under discussion, but according to the most popular hypothesis, the explosion occurs in a white dwarf star which has a binary companion.
A white dwarf star is the result of one of the possible ways that a star can end its life: in the form of a very compact star consisting mainly of carbon and oxygen which are the end products from the nuclear fusion processes taking place in the final phase of a star’s life. If a white dwarf is part of a binary star system (two stars orbitting a common center of mass), the white dwarf may start accreating material from the other star. At a certain point, the increased pressure and temperature from the accreted material may reignite fusion processes in the core of the white dwarf. This is the cause of the explosion. We will again defer details about the process to later lectures.
It can be shown that this explosion occurs when the mass of the white dwarf is close to the so-called Chandrasekhar limit which is about 1.4M . Since the mass of the exploding star is always very similar, the luminosity of the explosions will also be very similar. The absolute magnitude of a type Ia supernova is MV ≈ MB ≈= −19.3 with a spread of about 0.3 magnitudes. A more exact estimate of the absolute magnitude of a supernova may be obtained by the light curve. After reaching maximum magnitude, the supernova fades off during days, weeks or months. By observing the rate at which the supernova fades, one can determine the absolute magnitude of the supernova at its brightest.
Again, here we will only use the fact that the absolute magnitude of type
Ia supernovae can be obtained from its light curve in order to determine distances. More details about the physical processes giving rise to the explosion and to the fact that the light curve can be used to obtain the luminosity will be presented in later lectures. Supernovae can be used to determine distances to galaxies beyond 1000 Mpc.
3.4 The Tully-Fisher relation
The Tully-Fisher relation is a relation between the width of the 21 cm line of a galaxy and its absolute magnitude. As we remember, the 21 cm radiation is radiation from neutral hydrogen (look back at the lecture on electromagnetic radiation). Spiral galaxies have large quantities of neutral hydrogen and therefore emit 21 cm radiation from the whole disc. The 21 cm line is wide because of Doppler shifts: Hydrogen gas at different distances from the center of the galaxy orbits the center at different speeds giving rise to several different Doppler shifts. We also remember that the rotation curve for galaxies towards the edge of the galaxy was flat. So, gas clouds orbiting the galactic center at large distances all have the same orbital velocity vmax and thus the same Doppler shift. There are therefore many more gas clouds with velocity vmax than with any other velocity.
The flux at the wavelength corresponding to the Doppler shift
∆λmax
λ0 = vmax
c ,
is therefore larger than for instance at a wavelength of 21 cm itself. The result is a peak in the flux of the spectral line at either side of 21 cm at the wavelength 21 ± ∆λmax cm. The wavelength of this peak is a measure of the maximal velocity in the rotation curve:
vmax= c∆λmax λ0 .
We have seen in a previous lecture that the maximum velocity is related to the total mass of the galaxy. The higher the maximum velocity, the higher the mass (why?). If we assume that a higher total mass also means a higher content of lumious matter and therefore a higher luminosity, it is not difficult to immagine that a relation can be found between the maxi-mal speed measured from the 21 cm line and the luminosity, or absolute magnitude of the galaxy. The relation can be written as
MB = C1log10vmax+ C2,
where MB is the absolute magnitude at blue wavelenghts and C1 and C2 are constants depending on the type of spiral galaxy. The constant C1 is normally in the range −9 to −10 and C2 in the range 2.7 to 3.3. The Tully-Fisher relation can be used as a distance indicator out to distances beyond 100 Mpc.
3.5 Other distance indicators
Some other distance indicators:
• The globular cluster luminosity function: Globular clusters are clus-ters of stars containing a few 100 000 stars. These clusclus-ters are usu-ally orbiting a galaxy. A galaxy has typicusu-ally a few hundred globular clusters orbiting. It has been found that the luminosity function, i.e.
the percentage of globular clusters with a given luminosity, is simi-lar for all galaxies. By finding this luminosity function for galaxies with a known distance, the globular clusters can be used as distance indicators for other galaxies.
• The planetary nebula luminosity function: Planetary nebulae (which have nothing to do with planets) are clouds composed of gas which dying stars ejected at the end of their lifetime. The planetary nebu-lae have a known luminosity function which can be used as distance indicators for distant galaxies.
• The brightest galaxies in clusters: It has been found that the bright-est galaxies in clusters of galaxies have a very similar absolute magni-tude in all clusters. They can therefore be used as distance indicators to clusters of galaxies.